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In the following problem, concerning the stress vectors applied on the surface of the circle, why is there a minus sign in $t(- ê_r) = t(− \cos θ\ ê_1 − \sin θ\ ê_2)$?

In this problem, loads are applied as can be seen:

The solution from my course indicates the following answer:

Stress vector on the boundary of the circular cavity:
$t(- ê_r) = t(−\cos θ\ ê_1 − \sin θ\ ê_2) = 2.5(\cos θ\ ê_1 + \sin θ\ ê_2)$.

I just don't understand why there's a minus sign in the first two terms.

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The normal vector is in the negative direction of the unit vector for the radial direction in polar coordinates. That is,

$$\hat{n} = - \hat{\mathbf{e}}_r\, .$$

This is the reason for the minus sign.

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    $\begingroup$ I also found out that $\hat{n}$ is a vector pointing outward from the volume (in this exercise, the volume being is represented in gray, and the white part inside the circle in is the exterior of the volume, so this makes sense). $\endgroup$
    – c.leblanc
    Commented May 1, 2023 at 20:32

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